Question

in Matlab, Use the Monte Carlo analysis to compute the area of a circle with radius 1. print out your code, at least one figure on which the circle and ‘dart hits’ are shown, and numerical results for N=10, 100,1000. For each N, repeat the calculation at least 5times.

Answer 1

Matlab code for Monte Carlo simulation for finding area of circle clear all close all Radius of the circle % number of points generated N10 b-r x a (b-a).randN, 1)i y = a + (b-a ) . * rand ( N , 1); radii sqrt(x."2+y."2); hits sum(radiic-1 th-linspace (0,2*pi)i x r-cos (th)i y r-sin(th)i figure(1) plot(X-ty-r, . Linewidth' , 2) Plotting data k-0; hold on for i-1:N radd-sqrt((x(i))."2+(y(1)),"2); if radd<-1 k-k+1 plot ( x( i),y(i),'r*') else plot (x(i) ,y(i), end end xlim(t-1 11) ylim-1 11) title (sprintf('Monte Carlo plot for N-tdN)) xlabel( Ylabel'y' ar-4 hits/N; fprintf("\n\tFor N-td area of Circle using Monte Carlo method is n,N, ar)

clear all Radius of the circle r-1: % number of points generated N = 100;

x = a + (b-a ) . * rand N , 1 ) ; y-a (b-a).randN,1); radii - sqrt(x."2+y."2); hits sum4 rad11<=1 } ; th=linspace(0,2-pi); x r-cos (th); y r-sin(th)i figure (2) plot (x r,y r, Linewidth 2) Plotting data hold on for i-1:N if radd<-1 k k+1 else plot(x(i),yi), end end xlim(-1 11) ylim(-1 11 title(sprintf('Monte Carlo plot for N-dN)) xlabel ( 'x' ) ylabel('y' ar-4 hits/N; fprintf("\n\tpor Nold rea of Circle using Monte Carlo method is n,N, ar)

clear all もRadius of the circle r-1i t number of points generated N 1000 ar b = r; xa(b-a) rand(N,1)1 ya (b-a). rand (N,1) hits- surt radi i<=1); th linspace (0,2*pi) x r-cos (th) y r-sin(th)

figure( 3) plot (x r,y r, Linewidth',2) Plotting data k-0 hold orn for i=1:1 if radd<-1 xx(k)랭(i); YY(k) yi) plot(x(i),y(i),'r*') else plot(x y) end end xlim(-1 111 ylim(-1 1]) title { sprintf("Monte Carlo plot for 11-3d,,N)) xlabel('s) ylabel(y' ar-4*hits/N fprintf(,\n\tFor N d area of Circle using Monte Carlo method is In,N,ar) Por N-10 area of Circle using Monte Carlo method is 3.200000 For r=100 area of circle using Nonte Carlo method is 3.200000. Por N-1000 area of circle using Monte Carlo method is 3.156000

Monte Carlo plot for N 10 0.8 06 0.4 0.2 0.2 0.4 0.6 -0.8 1 08 6 04 204 08 08

Monte Carlo plot for N-100 08 0.6 0.4 0.2 24 0.2 -0.6 0.8 O.4 1 080.60402 02 040 08

Monte Cario plot for N 1000 参" + 04 1 -0806 0402 02 04 08 O8

%%Matlab code for Monte Carlo simulation for finding area of circle
clear all
close all
%Radius of the circle
r=1;
% number of points generated
N = 10;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(1)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
          
clear all

%Radius of the circle
r=1;
% number of points generated
N = 100;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(2)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
          
clear all
%Radius of the circle
r=1;
% number of points generated
N = 1000;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(3)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%